The application of inerter in tuned mass absorber

P. Brzeski a, E. Pavlovskaia b, T. Kapitaniak a, P. Perlikowski a,n

a Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Polandb Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, AB24 3UE, Aberdeen, Scotland

a r t i c l e i n f o

Article history:Received 11 March 2014Accepted 8 October 2014Available online 19 October 2014

Keywords:InerterDuffing oscillatorTuned mass absorberPendulumBifurcation analysisOptimization

a b s t r a c t

In this paper we investigate the dynamics of tuned mass absorber with additional viscous damper andinerter attached to the pendulum. The devices are used to damp out oscillations of non-linear Duffingoscillator. Analysis of how these devices influence the dynamics of tuned mass absorber and its dampingproperties is shown. We calculate the detailed bifurcation diagrams and show how by changing theparameters of damper and inerter one can eliminate dangerous dynamic instabilities from the systems.Finally, in the last section we present an optimization of TMA's parameters in order to achieve bestefficiency in damping of the main body vibrations.

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1. Introduction

The effective damping of mechanical and structural systemsoscillations has always been a big challenge for engineers. One ofthe first attempts to absorb energy of vibrations and in conse-quence reduce the amplitude of motion is a tuned mass damper(TMD) introduced by Frahm [1]. The device consists of mass onlinear spring such that its natural frequency is identical with thenatural frequency of damped system. As it is well known, classicTMD is extremely effective in reducing response of the mainstructure in its resonance but for other frequencies (even closeto this resonant frequency) it increases the amplitude of thesystem's motion.

Modification of TMD has been presented in the work of DenHartog [2]. The author proposes the addition of the viscous damperto Frahm's system design. Thanks to introduced damping TMDbecome a powerful device that can reduce vibrations of the mainbody in a wide range of excitation frequencies around principalresonance. Another modification that can lead to broaden the rangeof TMD effectiveness has been proposed by Roberstson [3] andArnold [4], who interchange linear spring of the TMD by the non-linear one (with linear and non-linear parts of stiffness). In recentyears much more attention has been paid to the possibility of usingpurely non-linear springs [5–7]. Authors show that systems withsuch springs have no main resonant frequency, hence the TMDworks in a wide range of excitation frequencies.

One can find many successful applications of TMDs which are usedto prevent damage of buildings due to seismic excitation [8,9],suppress vibration of tall buildings subjected to wind [10,11], achievethe best properties of cutting processes [12,13], decrease vibration offloors or balconies [14,15], reach stable rotations of rotors [16–18],stabilize drilling strings [19] and many others.

Another important contribution into subject of vibrations absorp-tion is an introduction of the device called a tuned mass absorber(TMA) proposed by Hatwal et al. [20–22] who interchange the linear/non-linear oscillator by the pendulum. In case of pendulum thenatural frequency depends only on its length, so TMA it much easierto tune in practical applications. Moreover, the disadvantage of TMD isthat it works only along its mounting direction. This drawback is notpresent for the TMA because pendulum can oscillate regardless of thedirection of base structure motion (in horizontal case pendulumoscillates for any excitation frequency while for vertical orientationonly in its parametric resonances). The dynamics of the TMA withvertical forcing of base oscillator is considered in a few papers [21–31].Presented analysis allows to understand the dynamics and response ofthe main structures around primary and secondary resonances of thependulum. The complete bifurcation analysis of the TMA applied tovertically forced Duffing oscillator in two parameters space (amplitudeand frequency of excitation) is presented in [32]. The similar analysis isalso performed for systems with horizontally forced main masses[33–36].

In our previous paper [37] we show the detailed analysis ofTMA with additional damper in the pivot of pendulum. We showthat properly chosen value of pendulum's damping coefficient canstrongly decrease the amplitude of Duffing oscillator aroundprimary resonance and broaden the range of TMA effectiveness.

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International Journal of Non-Linear Mechanics

http://dx.doi.org/10.1016/j.ijnonlinmec.2014.10.0130020-7462/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail address: przemyslaw.perlikowski@p.lodz.pl (P. Perlikowski).

International Journal of Non-Linear Mechanics 70 (2015) 20–29

We also present optimization scheme that can be used to determineoptimum values of damping parameters. In this paper we develop thisidea and consider systemwith not only additional damper but also theinerter. The inerter is a two terminal element which has the propertythat force generated at its ends is proportional to the relativeacceleration between them [38]. Smith proposed to use an inerter insuspensions of cars. He has shown that oscillations induced by roadimperfections and load disturbances can be reduced more effectivelyusing suspensions with inerters. In his further work [39] he hasstudied several simple passive suspension struts, each containing atmost one damper and inerter. The theoretical results are confirmed byexperiment showing that suspension layouts with inerter are moreeffective than classical designs with damper and springs only. In 2005the inerter has been profitably used as a part of suspension in Formula1 racing car [40]. Wang and Su [41] present how the performance ofsuspension is influenced by non-linearities which appear due to theinerter construction including effects caused by friction, backlash andelasticity. They show that the performance benefits are slightlydegraded by the inerter non-linearities but still the overall perfor-mance of suspension with a non-linear inerter is better than tradi-tional suspensions, especially when the stiffness of suspension is large.As one can expect successful application of an inerter in car suspen-sion resulted in a number of studies on other possible applications ofinerters. Takewaki et al. examine [42] if the advantages of inerter canbe beneficial in devices protecting buildings from earthquakes. Theauthors present detailed study showing how allocation of dampingdevice with inerter on each storey (from first to twelve storey)influences the response of the building. In a recent paper [43] authorsstudy the influence of inerter on the natural frequencies of vibrationsystems. They propose different constructional solutions of one andtwo degree-of-freedom systems and present how inerters influencetheir dynamics.

This paper is organized as follows: Section 2 contains descrip-tion of the considered model of the system. Section 3 is dividedinto three parts. In the first part we show the influence ofadditional viscous damper on performance of TMA, the secondpart is devoted to influence of inerter and in the last one wecompare how these devices affect the dynamics of the system.Finally, in Section 4, we consider the dynamics of the systemunder the presence of both damper and inerter and use optimiza-tion procedure to obtain the optimal values of additional devicesparameters which ensure the best damping properties. We sum-marize our results in Section 5.

2. Model of the system

We consider a horizontally forced single-well Duffing oscil-lator with attached T-shaped pendulum (Fig. 1(a)). Additional

devices – damper and inerter are mounted onto the base structureand attached to the ends of the arms of the T-shaped pendulum.The motion of the system is characterized by following generalizedcoordinates: the horizontal displacement of Duffing oscillator isdescribed by coordinate x and the angular displacement ofpendulum is given by angle φ. The notation of systems' para-meters is as follows: M is mass of Duffing oscillator, m and lcorrespond to mass and length of pendulum's rod, la is the lengthof the massless cross arms of T-shaped pendulum, k1 and k2 arelinear and non-linear parts of spring stiffness. The viscous damp-ing coefficient of Duffing oscillator is given by c and the viscousdamping in the pivot point of the pendulum by cP. Inertance of theinerter connected to the pendulum is given by parameter In whilecA is damping coefficient of additional viscous damper.

In Fig. 1(b) we present considered system with deflectedpendulum. In order to derive the equations of motion we haveto calculate the generated forces in the additional devices with thechange of angular position of the pendulum φ. The change in thelength of additional damper Δldamper can be calculated as follows:

Δldamper ¼ � la sinφcosαd

; ð1Þ

where la is the length of the pendulum's arm and αd is an angle bywhich the damper is deflected from the vertical position. Tocalculate the change in the length of inerter Δlinerter the followingformula can be used

Δlinerter ¼la sinφcosαi

; ð2Þ

where αi is an angle by which the inerter is deflected from thevertical position. As discussed in Appendix A the system can bedesigned to ensure that αd51 and αi51 which enables us tosimplify Eqs. (1) and (2):

Δldamper ¼ � la sinφ; ð3Þ

Δlinerter ¼ la sinφ; ð4Þ

hence:

Δ_ldamper ¼ � la _φ cosφ Δ€ldamper ¼ � la €φ cosφþ la _φ2 sinφ; ð5Þ

Δ_linerter ¼ la _φ cosφ Δ€linerter ¼ la €φ cosφ� la _φ2 sinφ: ð6Þ

Using the above calculated velocities we can derive the followingformula for the kinetic energy T, potential energy V, Rayleighdissipation D and general forces Q for the considered system:

T ¼ 12ðMþmÞ _x2þ1

6ml2 _φ2þ1

2ml _x _φ cosφþ1

2InðΔ_linerterÞ2 ð7Þ

M

c

1

lm

F ( t)cos ω

k +k x1 22

x

φ

I cn A

la

αi αd

φ

φb)

Fi

Fd

( )φT

Fig. 1. Model of the system (a) and orientation of forces generated by additional devices (b).

P. Brzeski et al. / International Journal of Non-Linear Mechanics 70 (2015) 20–29 21

V ¼ 12k1x2þ

14k2x4�

12mlg cosφ ð8Þ

D¼ 12c _x2þ1

2cAðΔ_ldamperÞ2 ð9Þ

Q ¼ Tð _φÞ∂φ∂φ

ð10Þ

where Tð _φÞ ¼ cP _φ is a damping momentum in the pivot point ofthe pendulum. Using Lagrange equations off the second type weget the equations of motion:

ðMþmÞ €xþ12ml €φ cosφ� _φ2 sinφ

h iþk1xþk2x3þc _x ¼ F cos ðωtÞ

ð11Þ

12ml €x cosφþ 1

3ml2þ Inl

2a cos

2φ� �

€φþ12mlg sinφ

þðcPþcAl2a cos

2φÞ _φ� Inl2a _φ

2 cosφ sinφ¼ 0 ð12Þ

Introducing dimensionless time τ¼ tω1, where ω1 ¼ffiffiffiffiffiffiffiffiffiffiffiffik1=M

pis the

linear approximation of natural frequency of Duffing oscillator weobtain dimensionless equations:

ð1þmDÞ ~x″þ12mDlD φ″ cosφ�ðφ0Þ2 sinφ

h iþ ~xþk2D ~x

3þcD ~x0 ¼ FD cos ð ~ωτÞ

ð13Þ

12mDlD ~x

″ cosφþ 13mDl

2Dþ InDl

2aD cos

2φ� �

φ″þ12mDlDgD sinφ

þðcPDþcADl2aD cos

2φÞφ0 � InDl2aDðφ0Þ2 cosφ sinφ¼ 0

ð14Þ

where prime denotes the differentiation with respect to non-dimensional time τ, ~x ¼ x=l0, ~x 0 ¼ _x=l0ω1, ~x″¼ €x=l0ω2

1, φ0 ¼ _φ=ω1,

φ″¼ €φ=ω21, (for simplicity tildes in dimensionless equations

will henceforth be omitted) k2D ¼ k2l20=Mω2

1, cD ¼ c=Mω1,cPD ¼ cP=Ml20ω1, cAD ¼ cA=Mω1, InD ¼ In=M, lD ¼ l=l0, laD ¼ la=l0,mD ¼m=M, gD ¼ g=l0ω2

1, FD ¼ F=Ml0ω21, ~ω ¼ω=ω1.

We take initial parameters values from our previous paper [37]which are as follows: M ¼ 3:63 ½kg�, k1 ¼ 660 ½N=m�, k2 ¼ 200 ½N=m�,c¼ 0:240 ½Ns=m�, cP ¼ 1:977 ½Nms�, l¼ 0:0779 ½m�, la ¼ 0:01 ½m�,m¼ 0:363 ½kg�, F ¼ 0:33 ½N�. Values of parameters cAD and InD whichare used to describe the additional damper and inerter respectivelywill be changed during numerical simulations in order to show howthese supplementary devices influence the efficiency of the TMA andthe dynamics of the considered system.

Using reference length l0 ¼ 1:0 ½m� we perform transformation todimensionless parameters (depicted by letter D) in a way that allowsto hold accessibility to physical parameters receiving: k2D ¼ 0:30303,cD ¼ 0:0049, cPD ¼ 4:04� 10�2, lD ¼ 0:0779, laD ¼ 0:01, mD ¼ 0:1,gD ¼ 0:053955, FD ¼ 0:0005.

3. Numerical results

3.1. Influence of additional damper on the response of the Duffingoscillator

In our previous paper [37] we have investigated the dynamicsof the horizontally forced Duffing oscillator with suspended TMAsof three different types, i.e., classical single pendulum, dualpendulum and pendulum-spring. We have shown how differentparameters of TMAs affect the behavior of the system and thenproposed an optimization method which can be used to adjust theabsorbers parameters to obtain the best damping properties in awide range of excitation frequencies. One of the main conclusionsdrawn from the analysis of single pendulum TMA is that dampingcoefficient in the pivot point of the pendulum has a decisiveinfluence on the device efficiency. When damping coefficient is toosmall one can observe the decrease of Duffing oscillator amplitudeonly around principal resonance and when it is too big, TMAbarely changes the response of the base structure. In real worldapplications it can be difficult to ensure specifically identifiedvalue of viscous damping coefficient in the pivot point of thependulum. Therefore, in this paper we propose the modification ofthe pendulum's design (by introducing T-shaped pendulum) thatallows the attachment of oil viscous damper or a magnetorheolo-gical damper. Thus, total damping of the pendulum consists of twocomponents: damping in the pivot point described by parametercPD ¼ 4:04� 10�2 which is equivalent to 1% of critical damping ofthe pendulum and damping introduced by additional damperdescribed by parameter cAD. In this case the total damping of thependulum can be easily controlled. To show dynamics of Duffingsystem under the presence of additional damper in detail wepresent how frequency response curve (FRC) changes for increas-ing value of cAD. To obtain the FRC we use the Auto 07p [44]continuation toolbox. In Fig. 2(a) we present FRCs calculated forthe system without additional damper (in this case we consideronly damping in the pivot point) and for the system withadditional viscous damper characterized by parametercPD ¼ 5:5� 10�2. In both cases we assume that there is no inerterattached to the arm of the pendulum (InD ¼ 0).

1.0 2.18.0 0.9 1.1

0.02

0.10

0

xam

)x(

0.04

0.06

0.08

NS

ω

c = 5.5 10AD-2

c = 0AD

0.862 0.864 0.866 0.868

NS

0.017

0.018

0.019(c)

(c)

SB

(b)

0.07

0.08

0.09

0.060.96 0.97 0.98 0.99

PD

(b)(a)

Fig. 2. FRCs of the base structure calculated for the system without inerter (InD ¼ 0): with no additional damper cAD ¼ 0, (continuous line) and with additional dampercAD ¼ 5:5� 10�2 (dashed line) showing how the increase of pendulum's total damping coefficient influences the response of the base structure. Subplots (b,c) aremagnifications of FRC curve presented in subplot (a). The stability of solutions is depicted by color of lines: black color corresponds to stable and red color to unstablesolutions. The damping in pivot of pendulum is present for both cases. (For interpretation of the references to color in this figure caption, the reader is referred to the webversion of this paper.)

P. Brzeski et al. / International Journal of Non-Linear Mechanics 70 (2015) 20–2922

Along the solid line presented in Fig. 2(a) that corres-ponds to the system without additional damper one can see thefollowing bifurcations: forω¼ 0:867907,ω¼ 0:861789,ω¼ 1:11276,ω¼ 0:980540 saddle-node bifurcations occur, for ω¼ 0:863156,ω¼ 0:868125 Neimark–Sacker bifurcations appear and for ω¼0:983657, ω¼ 0:9865495 symmetry breaking pitchfork bifurcationsoccur. The two asymmetric solutions created in pitchfork bifurcationlose their stability in a period doubling bifurcations that occur forω¼ 0:980868,ω¼ 0:981021. Bifurcations described above give rise tothe coexistence of different attractors which can be observed in awiderange of excitation frequency ω. Therefore, the dynamics of theconsidered system without additional damper is complicated andsuch a design is inconvenient for practical applications. Dashed linepresented in Fig. 2(a) is calculated for the system with additionaldamper characterized by parameter cAD ¼ 5:5� 10�2. Due to thepresence of supplementary damper, total damping coefficient of thependulum rises causing the decrease of Duffing amplitude andelimination of all bifurcations. To present how the position of eachbifurcation on the FRC changes with increase of the dampingcoefficient of additional damper, we calculated two parametersbifurcation diagram (ω, cAD) presented in Fig. 3. Analyzing bifurcationcurves presented in Fig. 3 one can see that Neimark–Sacker bifurca-tions disappear if cAD40:656� 10�2 (which corresponds to about0.16% of critical damping), and symmetry breaking bifurcations vanishfor cAD41:214� 10�2 (0.3% of critical damping). Therefore if damp-ing coefficient of additional damper is greater than 0.3% of criticaldamping one will observe only symmetric periodic solutions of thesystem. To eliminate all bifurcations and receive FRC without anycoexistence of attractors one has to ensure that cAD45:196� 10�2

meaning that additional damper damping coefficient is greater than1.29% of critical damping. All above values are relatively small. More-over, as we proved in our previous publication [37] for system withgiven parameters best damping efficiency of TMA can be achieved iftotal damping coefficient of the pendulum is equal to about 18% ofcritical damping. Therefore if additional damper damping coefficient isclose to the optimal value it will be effective with both suppressing thesystem's response as well as elimination of dangerous dynamicalphenomena. We do not show period doubling bifurcations of asym-metric solutions because they disappear simultaneously with pitchforkbifurcations.

3.2. Influence of the inerter on the response of the Duffing oscillator

As we mentioned before, there are no previous studies onapplication of inerter in TMA. Hence, the influence of the inerter

on damping properties of TMAs needs to be carefully examined.Similar to previous paragraph, to show how the dynamics of Duffingsystem changes under the presence of inerter, FRCs are used. Suchapproach enables accurate description of how the value of additionalinertance attached to the T-shaped pendulum (InD) affects dynamicalresponse of the system and damping properties of TMA. In Fig. 4(a) we show two FRCs, solid line corresponds to the system withoutthe inerter, and dashed one to system with additional inerterdescribed by dimensionless parameter InD ¼ 22:5. In both cases weassume that there is no additional damper (cAD ¼ 0) thereforedamping is present only in the pivot of the pendulum and it is equalto 1% of critical damping (cPD ¼ 4:04� 10�2). Black line presented inFig. 4(a) is identical to the one presented in Fig. 2(a) and should beused as a reference. Analysis of dashed FRC presented in Fig. 4(a) proves that by addition of the inerter one can eliminate allbifurcations that occur along the FRC of base structure. Simulta-neously, we see that the influences of inerter and additional damperon the systems dynamics is different.

Two parameters bifurcation diagram (ω, InD) shown on Fig. 5details the influence of the additional inertance on position of eachbifurcation along the FRC. For the system without inerter – regardlessof total pendulum damping coefficient – all bifurcations occur in therange ωAð0:85; 1:15Þ: Addition of inerter causes the shift in thebifurcations position towards the smaller values of forcing frequency.Therefore, the range of parameterω for which we can observe saddle-node bifurcations rapidly moves towards smaller values of ω. Analyz-ing bifurcation curves shown in Fig. 5 one can see that Neimark–Sacker bifurcations disappear if InD40:143, and symmetry breakingbifurcations vanish for InD41:436. This means that even for thesystemwithout additional damper we can eliminate non-periodic andnon-symmetric solutions by the introduction of the inerter. Unfortu-nately, the elimination of all bifurcations requires relatively highinertance values, and it can be achieved for InD420 which meansthat inerter has 20 times bigger inertia than the Duffing oscillator.Such device can be easily constructed by the use of multi-stage gearunit but it would completely mitigate pendulum's motion andeliminate the damping properties of TMA (see FRC calculated forInD ¼ 22:5 presented in Fig. 4(a)). However, it is important to noticethat the range of coexistence of different solutions caused by saddle-node bifurcations is extremely small if InD44.

3.3. Comparison of the inerter and the damper influences

In Sections 3.1 and 3.2 we presented how the damper and theinerter influence the dynamical response of the Duffing oscillator.

0

1 10

cAD

ω

-2

2 10-2

3 10-2

4 10-2

5 10-2

2.11.10.19.08.0

Symmetry breaking bifurcationNeimark-Sacker bifurcation

Saddle-node bifurcation

Fig. 3. Two parameters bifurcation diagram (ω and cAD) showing how the value of additional damper damping coefficient affects the position of bifurcation points on theFRCs. The damping in pivot of pendulum is present.

P. Brzeski et al. / International Journal of Non-Linear Mechanics 70 (2015) 20–29 23

In this section we compare the effects caused by these devices.Results presented in Figs. 3 and 5 can be used to correlate howadditional damper and inerter change the response of basestructure by shifting the position of the bifurcations. In bothFigs. 3 and 5 bottom horizontal line of the graph refers to lack ofthese devices (cAD ¼ 0, InD ¼ 0). The FRC for such case is presentedboth in Figs. 2(a) and 4(a) and is used as a reference line. When weintroduce the additional damper to the system, we see that even ifits damping coefficient is relatively small it is easy to eliminate allbifurcations. If we want to use only an inerter to eliminate allbifurcations it should have comparatively high inertance value.Despite the value of parameter cAD all bifurcations occur in rangeωA ð0:85; 1:15Þ but the increase of inertance parameter InD leadsto extension of this range towards smaller values of forcingfrequency (ωAð0:31; 1:12Þ). When the damping coefficient riseswe initially observe rejection of bifurcations that occur around firstresonance peak (ω� 0:87). If cAD42� 10�2 we observe onlysaddle-node bifurcations (two or four) around second resonancepeak (ω� 0:87). Contrary, with increase of additional inertance wefirstly observe elimination of bifurcations that occur aroundsecond resonance peak. Two saddle-node bifurcations endure upto InD ¼ 20 but distance between them is extremely small and withthe increase of InD their position shifts towards smaller values of ω(up to ω¼ 0:3).

To further analyze the difference between the effects that arecaused by inerter and damper we calculated six FRCs of considered

system that are presented in Fig. 6. Three of them correspond tothe system with different inerters attached only, and other threeshows the response of the system in a presence of differentadditional dampers only. In the upper row we show the responseof Duffing oscillator, while in the lower row the maximumamplitude of pendulum versus frequency of excitation ω.

First two subplots of Fig. 6 presents the changes in the dynamicresponse of Duffing oscillator. Analyzing Fig. 6(a) one can say thatincrease of additional inertance leads to the following effects: firstresonance peak becomes much smaller while second one is slowlyrising. Both peaks occur for the smaller ω as InD increases. If wesubstitute inerter with damper (see Fig. 6(b)) and increase dampingcoefficient we notice that two resonance peaks merge into one thatarise around ω¼ 0:95. But the most essential effect of totaldamping coefficient increase is the significant reduction of thepeak height which cannot be observed in system with inerter only.Hence, while inerters can be used for elimination of bifurcations,they do not affect damping properties positively. Subplots (c,d) inFig. 6 refer to the response of the pendulum. Analysis of Fig. 6(d) leads to conclusion that by addition of damper which is attachedto the arm of T-shaped pendulumwe can easily decrease its motionamplitude in a wide range of forcing frequencies. Moreover, thiseffect is present even for relatively small damping coefficients (seedashed curve presented in Fig. 6(d) calculated for cAD ¼ 0:404 whichcorresponds to 10% of critical damping). If we want to suppresspendulum's motion with comparable effectiveness we have to use

0

5

10

15

20

(b)

0.3

InD

ω0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

(a)

0.4

0.84 0.85 0.860.830

0.2

0.6

0.3

0.976

0.6

0.9

1.2

1.5

00.980 0.984

Symmetry breaking bifurcationNeimark-Sacker bifurcation

Saddle-node bifurcation

Fig. 5. Two parameters bifurcation diagram (ω and InD) showing how the value of additional inertance affects the position of bifurcation points on the FRCs. The damping inpivot of pendulum is present.

1.0 2.18.0 0.9 1.1

0.02

0.10

0

xam

)x(0.04

0.06

0.08

ω

without inerterI =22.5n

NS

0.862 0.864 0.866 0.868

NS

0.017

0.018

0.019

SB0.07

0.08

0.09

0.060.96 0.97 0.98 0.99

PD

(c)

(c)

(b)(b)

(a)

Fig. 4. FRCs of the base structure calculated for system without inerter and additional damper (cAD ¼ 0, InD ¼ 0) (solid line) and for the system with inerter only InD ¼ 22:5(dashed line) showing how the addition of inerter influences the response of the base structure. Subplots (b,c) are magnifications of FRC curve presented in subplot (a). Thestability of solutions is depicted by color of lines: black color corresponds to stable and red color to unstable solutions. The damping in pivot of pendulum is present for bothcases. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

P. Brzeski et al. / International Journal of Non-Linear Mechanics 70 (2015) 20–2924

inerter with comparatively high inertance (see dotted line pre-sented in Fig. 6(c) calculated for InD ¼ 22:5). Otherwise, with theincrease of parameter InD we observe slow reduction in peaksheight and their migration toward smaller forcing frequencies.

4. Optimization of additional devices parameters

Up to this moment we considered systems with only oneadditional device (damper or inerter). Such approach allow us tocompare the effects caused by both devices and to emphasizedifferences between their influence on systems dynamics. Weclaim that a combination of both inerter and additional dampercan lead to boost in TMA's damping efficiency. In this section wevalidate this assumption and present how the response of basestructure changes for different values of parameters InD and cAD.

In order to visualize the behavior of the system with bothinerter and additional damper in Fig. 7(a,b,e,f) we present threedimensional plots created as a surfaces based on the FRCs of thesystem. To give better overview of the Duffing system amplitudewe also present two dimensional maps which are flat projectionsof three dimensional surfaces presented in subplots (c,d,g,h) ofFig. 7. For each plot we assume different additional dampercoefficient and present how inertance value influences responseof Duffing oscillator changing InD from 0 to 1. In this section weconfine considered range of InD because TMA damping efficiencydeteriorate for higher values of InD. Plots (a,c) were calculated forcAD ¼ 0:202 which equals to 5% of critical damping, plots (b,d) forcAD ¼ 0:404 which stands for 10% of pendulum critical damping,subplots (e,g) represent the response of the system for cAD ¼ 0:808(20% of critical damping) and (f,h) for cAD ¼ 1:212 (30% of criticaldamping). Analyzing Fig. 7 one can say that for every consideredvalue of damping coefficient one can achieve the decrease ofDuffing oscillator amplitude by addition of proper inerter. This

effect is highly visible for smaller values of cAD when the additionof inerter helps to improve TMA's efficiency significantly for a widerange of ω.

Analyzing the results obtained for cAD ¼ 0:202 (Fig. 7(a,c)) wesee that for system without inerter first resonance peak whichoccurs around ω¼ 0:85 is dominant. If we introduce inerter andincrease parameter InD we see that the height of first resonancepeak decreases while second resonance peak is growing andbecomes dominant for InD40:5. Similar phenomena can beobserved for the system with increased additional dampingcoefficient cAD ¼ 0:404 (Fig. 7(a,c)). The only difference is thatbecause total damping coefficient of the pendulum is closer tooptimal value both resonance peaks are smaller and less slender.If we increase additional damper damping coefficient up to 20% ofcritical damping (cAD ¼ 0:808) we observe only one resonancepeak (Fig. 7(e,g)). For the system without inerter, it occurs aroundω¼ 0:9. If we introduce inerter and increase supplementaryinertance InD, we also observe migration of the peak's positiontowards higher values of excitation frequencies. Decrease in theamplitude of base structure can be observed for InD � 0:5 but it isnot so significant as for smaller values of cAD. Effects caused by aninerter are similar to cAD ¼ 1:212 (Fig. 7(f,h)) but for this caseimprovement in damping efficiency is barely visible. Therefore, iftotal damping coefficient of the pendulum is relatively large (inour case greater than 30% of critical damping) it suppress pendu-lum's vibrations so much that inerter's influence on its dynamics isbarely visible.

Comparing results obtained for damping coefficients equal to10% and 20% of critical damping, we see that comparable dampingefficiency can be achieved for different composition of parameterscAD and InD. Still, by changing the configuration of differentdampers and inerters, we can better adjust the shape of FRC toour expectations. Moreover, in many cases inerters can be easierand more convenient to apply precise inertance value.

ω1.00.8 1.2 1.40.6 1.60.4

0.02

0.10

0

xam

)x(0.04

0.06

0.08

ω1.00.8 1.2 1.40.6 1.60.4

0

xam

)(φ

π

π

π

π

18

14

38

12

ω1.00.8 1.2 1.40.6 1.60.4

0.02

0.10

0

x am

) x(

0.04

0.06

0.08

ω1.00.8 1.2 1.40.6 1.60.4

0

xam

)( φ

π

π

π

π

18

14

38

12

Fig. 6. Comparison between systems with attached inerter and additional damper. (a,b) FRCs of Duffing oscillator and (c,d) FRCs of pendulum calculated for systems withthree different inerters (a,c), and three different dampers (b,d). For plots in each column parameters of the systems are identical.

P. Brzeski et al. / International Journal of Non-Linear Mechanics 70 (2015) 20–29 25

ω

0.2

0

0.4

0.6

0.8

1.0

1.00.8 1.2 1.40.6

InD)x(

xam

ω

InD

ω

InD

ω

0.2

0

0.4

0.6

0.8

1.0

1.00.8 1.2 1.40.6

InD

ω

0.2

0

0.4

0.6

0.8

1.0

1.00.8 1.2 1.40.6

InD

ω

InD

ω

0.2

0

0.4

0.6

0.8

1.0

1.00.8 1.2 1.40.6

InD

ω

InD

)x(xa

m

)x(xa

m

)x(xa

m

Fig. 7. Three dimensional plots presenting FRC of the systemwith inerter and additional damper (a, b, e, f) and two dimensional maps (c, d, g, h) which are flat presentationsof subplots (a, b, e, f) respectively. Plots (a, c) were calculated for cAD ¼ 0:202 (5% of critical damping), plots (b, d) for cAD ¼ 0:404 (10% of critical damping) and plots (e, g) forcAD ¼ 0:808 (20% of critical damping) and (f, h) for cAD ¼ 1:212 (30% of critical damping).

P. Brzeski et al. / International Journal of Non-Linear Mechanics 70 (2015) 20–2926

5. Conclusions

In this paper we examine how additional devices attached tothe T-shaped pendulum TMA influences the dynamical response ofbase structure. In the beginning we propose change in thependulum's shape and analyze its consequences. Introduction ofthe T-shaped pendulum enables easier control of total dampingcoefficient of the pendulum and allows the addition of othersupplementary devices that can modify the response of thesystem. Simultaneously, changes in the TMA design can be con-ducted in a way that would not cause undue complexity of themodel. Therefore, we claim that proper modification of pendu-lum's shape can significantly facilitate process of adjusting TMAperformance to required level.

In Section 3 we investigated how additional damping andinertance introduced by supplementary devices influence thesystem dynamics and compare the effects caused by additionaldamper and inerter. Addition of both components can lead toelimination of bifurcations that occur for the systemwithout them.In the considered case, if we do not want to observe anybifurcations we should add damper with damping coefficientcorresponding to 1.29% or more of critical damping or an inerterwith 20 or more times bigger inertia than Duffing oscillator.Therefore, dampers are much more effective in cancelingunwanted bifurcations. Moreover, addition of proper damper leadsto increase of TMA's damping efficiency in a wide range ofexcitation frequencies while inerter can cause only slightlydecrease of base structure amplitude. Simultaneously, combina-tion of these two devices can lead to significant improvement ofdamping properties and enables much better control on the shapeof FRCs. Therefore we can precisely adjust the response ofconsidered structure to our expectations by selecting the rightconfiguration of damper and inerter. In Section 4 we picked fourdifferent additional dampers and – for each – presented how thechange of the inertance value influences response of Duffingoscillator. For every considered value of damping coefficient,decrease of the Duffing oscillator amplitude can be achieved byaddition of a proper inerter. In the considered system, introductionof the inerter which has around half of Duffing oscillator inertiaprovides best damping properties despite the value of dampingcoefficient. Considering the fact that it is easier to obtain constantand precise value of inertance than damping coefficient, in manycases the combination of inerter and damper can be much moreconvenient to apply and enables better control of systemsdynamics.

Acknowledgment

This work has been supported by Young Scientists Fund ofFaculty of Mechanical Engineering at the Lodz University of Lodz.

Appendix A

In this section we consider the geometry of the system in detailas shown in Fig. 8.

To calculate forces generated by additional damper and inerterthat are coupled to the arms of T-shaped pendulum we have toknow the actual deformations of the devices. In order to deriveformulas that describes how lengths of the devices change withthe change in deflection of the pendulum we have to usegeometrical measures of the system which are presented inFig. 8. The angular displacements of pendulum are given by angleφ, the length of the arms of T-shaped pendulum is given by la,initial lengths of both devices are the same and described by

parameter ld. Actual length of expanding device will be describedby l1 and angle of its inclination by α1. For contracting device wewill use l2, and α2 respectively. Actual lengths of the devices aregiven by the following formulas:

l1 ¼ld

cosα1þ la sinφ

cosα1ð15Þ

l2 ¼ld

cosα2� la sinφ

cosα2ð16Þ

Hence

l1 cosα1 ¼ ldþ la sinφ ð17Þ

l2 cosα2 ¼ ld� la sinφ ð18ÞTherefore we can get the following formulas:

α1 ¼ arccosldþ la sinφ

l1

� �ð19Þ

α2 ¼ arccosld� la sinφ

l2

� �ð20Þ

To describe how design of the system influences the maximumvalues of angles α1 and α2 we introduce the following ratio:

lratio ¼ldla

ð21Þ

Next we can determine actual lengths of the devices with thefollowing formulas:

l1 ¼ laffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ l2ratio�2 cosφþ2lratio sinφ

qð22Þ

l2 ¼ laffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ l2ratio�2 cosφ�2lratio sinφ

qð23Þ

Substituting formulas (22), (23) (into 19) and (20) we obtain

α1 ¼ arccoslratioþ sinφffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2þ l2ratio�2 cosφþ2lratio sinφq

0B@

1CA ð24Þ

α2 ¼ arccoslratio� sinφffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2þ l2ratio�2 cosφ�2lratio sinφq

0B@

1CA ð25Þ

In this paper we assume that in the described system, theangular displacement of pendulum is never greater than π=2(φrπ=2). Therefore in our case the maximum values of anglesα1 and α2 can be observed for φ¼ π=2. In Fig. 9(a) we present how

Fig. 8. Geometric measures that are used to describe actual lengths of devices.

P. Brzeski et al. / International Journal of Non-Linear Mechanics 70 (2015) 20–29 27

maximum values of angles α1 and α2 changes with the change oflratio when φ¼ π=2.

During optimization procedures we change value of inertanceand damping coefficient of additional damper. Both of thesedevices cause a decrease in the amplitude of pendulum's motion.For chosen amplitude of excitation if damping coefficient is largerthan 2% of critical damping, we do not observe angular deflectionsof pendulum greater than π=4 (φoπ=4). Hence in Fig. 9(b) wepresent how lratio influences the maximum values of angles α1 andα2 when φ¼ π=4. One can see that for lratio44 both α1 and α2 arealways smaller than 51.

Analyzing formulas (24) and (25), one can see that values ofangles α1 and α2 depend on two values: lratio which is constant anddetermined by the design of the system and angular displacementof the pendulum given by coordinate φ. In Fig. 10 we show howangle of pendulum's inclination influences angles α1 and α2 forfour different values of lratio (lratio ¼ 2, lratio ¼ 4, lratio ¼ 6, lratio ¼ 10).

In our analysis we assume that lratio ¼ 6, therefore α1 and α2 canbe treated as small angles. Due to this assumption we canintroduce the following simplifications:

αi51sinαiCαi cosαiC1sin ðφ7αiÞC sinφ cos ðφ7αiÞC cosφ

for i¼ 1;2 ð26Þ

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5 10 15 20

0.1

0.2

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10

15

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10 0

lratio

]dar[α

α1α2

5 10 15 20

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0.1

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Fig. 10. Graphs showing how angles α1 (a) and α2 (b) changes with the change of angular displacement of the pendulum φ for different values of lratio parameter.

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